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The path-groupoid $\mathcal{P}_1(X)$ of a (smooth) topological space $X$ is a refinement of the fundamental groupoid $\Pi_1(X)$ whose morphisms are given by (piecewise smooth) paths in $X$ up to thin-homotopy (rather than full homotopy). Thin-homotopies are basically homotopies sweeping zero area. (Note that there's a natural map $\mathcal{P}_1(X)\to \Pi_1(X)$ by sending thin-homotopies to full-homotopies.)

When we have a topological stratified space $(X,S)$, we may consider a different refinement of the fundamental groupoid $\Pi_1(X)$ given by the exit-path category $\text{Exit}(X,S)$. In particular, instead of considering the full set of paths, we look at the set of exit-paths: a path $\gamma:[0,1]\to X$ is an exit-path with respect to the stratification $S$ if, for each $t_1\leq t_2\in [0,1]$, the dimension of the stratum containing $\gamma(t_1)$ is less than or equal to the dimension of the stratum containing $t_2$. That is, exit-paths go up the strata. The morphisms of the exit-path category $\text{Exit}(X,S)$ are then given by exit-paths modulo (full) homotopy. (Note that $\Pi_1(X)\hookrightarrow EP(X,S)$ with respect to the trivial stratification.)

There are two sheaf-theoretic characterizations of the categories $\Pi_1(X)$, $\text{Exit}(X,S)$ given by corresponding equivalences of categories, namely:

  • the category of representations of the fundamental groupoid $\Pi_1(X)$ is equivalent to the category of local systems (say, for $X$ locally simply connected); see, for instance, section 2.6 Szamuely;

  • the category of representations of the exit-path category $\text{Exit}(X,S)$ is equivalent to the category of constructible sheaves; due to MacPherson, see Treumann; also mathoverflow.

Question

Is there an analogous sheaf-theoretic characterization of the category of representations of the path-groupoid $\mathcal{P}_1(X)$?

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Carlos
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    Is the guess maybe "bundles equipped with a (not necessarily flat) connection"? But I don't think I understand what a 'thin homotopy' is well enough to stand by that claim. – Dylan Wilson Sep 27 '17 at 19:29
  • Which category are you asking for representations in? Vect? If so, then Dylan has it right. Similarly if reps are in GTors for G a Lie group (this is work of Schreiber and Waldorf) – David Roberts Sep 27 '17 at 21:33
  • @DavidRoberts I was thinking about representations in Vect. But, for starters, Sets would do. I am indeed aware of Schreiber & Waldorf's work (link) where they explore the so-called transport functors. I am under the impression that these only give a particular sector of the category of representations of $\mathcal{P}_1(X)$ satisfying further descent data, therefore missing "all the others". Correct me, please, if I'm wrong. – Carlos Sep 28 '17 at 00:05
  • As @DylanWilson guessed quite well, these functors encode the information of the parallel transport on bundles with (not necessarily flat) connection. And in a way, we could say that the category of representations of the path-groupoid is equivalent to the category of bundles with (not necessarily flat) connection. – Carlos Sep 28 '17 at 00:06
  • However, I didn't mention this explicitly as I was trying to avoid influencing you going further into this direction. In the same way we have the web of correspondences {vector bundles with flat connection} ~ {representations of the fundamental groupoid} ~ {local systems}, we should(?) have a similar one when we lift flatness. The above is giving just one corner of this picture. What replaces local systems? It's not constructible sheaves as we know they give representations of the exit-path category (which can be further related to constructible vector bundles with flat connection). – Carlos Sep 28 '17 at 00:08
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    In addition to the above-cited work of Schreiber and Waldorf, I would like to advertise https://arxiv.org/abs/1501.00967, which removes some technical assumptions of Schreiber and Waldorf (such as sitting instants) and proves a similar result. – Dmitri Pavlov Sep 28 '17 at 23:09
  • Furthermore, in a forthcoming work with Stolz and Teichner I answer an n-dimensional version of your question: functors from the smooth path n-groupoid with thin homotopies as (n+1)-morphisms recover precisely bundles with n-connections. For n=1 one recovers the above results for vector and principal bundles with connection. For n>1 one recovers the n-category of bundle (n-1)-gerbes with connection. – Dmitri Pavlov Sep 28 '17 at 23:17
  • @DmitriPavlov Nice, thank you! In fact, it answers a "question" I had from the first comment, whether or not the category of representations of the path-groupoid encompasses more stuff than "just" vector bundles with connection. That is, any (smooth) functor $\mathcal{P}_1(X)\to \text{Vect}$ satisfies the appropriate descent data to assemble a vector bundle with connection (via parallel transport), and not just the transport functors from Schreiber & Waldorf. How are the results related? – Carlos Sep 29 '17 at 17:25
  • In any case, my question remains. What replaces local systems in this non-flat case? – Carlos Sep 29 '17 at 17:26

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